LAT: a Latinized aperiodic tiling for any sample size
Abstract
Quasi-Monte Carlo methods allow computer experiments to be run with far fewer simulations than crude Monte Carlo. A digital net in base two, such as Sobol', is however only balanced when the number of samples is a power of two, and Latin Hypercube Sampling (LHS) accepts any sample size but only controls the one-dimensional margins. This work proposes a space- filling design defined for any sample size, referred to as LAT for Latinized aperiodic tiling. The unit hypercube is cut recursively across its longest edge following the golden section into N cells of equal volume. One point is then placed in each cell, and the margins are made Latin while every point stays inside its own cell. The construction only uses integer splits and costs O(dNlog N). The recursion is shown to follow the Fibonacci word and its aperiodicity is analysed. LAT is assessed with four L2-discrepancies and with the integration error on analytical functions and engineering emulators, and compared to Monte Carlo, LHS, Halton, Sobol' and a rank-1 lattice. LAT is better than Monte Carlo and LHS as soon as the integrand has interactions. Sobol' remains more accurate at the powers of two, but its error is one to two orders of magnitude larger at other sample sizes. The accuracy of LAT does not depend on the sample size. Finally, the cells form a partition of the hypercube for any N. This allows one to refine the design locally, to search for an optimum by splitting cells and to sample non-rectangular regions.
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BibTeX
@misc{roy2026latinized,
title = {{LAT: a Latinized aperiodic tiling for any sample size}},
author = {Pamphile T. Roy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31593},
eprint = {2609.31593},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31593},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - LAT: a Latinized aperiodic tiling for any sample size
AU - Pamphile T. Roy
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31593
UR - https://arxiv.org/abs/2609.31593
AB - Quasi-Monte Carlo methods allow computer experiments to be run with far fewer simulations than crude Monte Carlo. A digital net in base two, such as Sobol', is however only balanced when the number of samples is a power of two, and Latin Hypercube Sampling (LHS) accepts any sample size but only controls the one-dimensional margins. This work proposes a space- filling design defined for any sample size, referred to as LAT for Latinized aperiodic tiling. The unit hypercube is cut recursively across its longest edge following the golden section into N cells of equal volume. One point is then placed in each cell, and the margins are made Latin while every point stays inside its own cell. The construction only uses integer splits and costs O(dNlog N). The recursion is shown to follow the Fibonacci word and its aperiodicity is analysed. LAT is assessed with four L2-discrepancies and with the integration error on analytical functions and engineering emulators, and compared to Monte Carlo, LHS, Halton, Sobol' and a rank-1 lattice. LAT is better than Monte Carlo and LHS as soon as the integrand has interactions. Sobol' remains more accurate at the powers of two, but its error is one to two orders of magnitude larger at other sample sizes. The accuracy of LAT does not depend on the sample size. Finally, the cells form a partition of the hypercube for any N. This allows one to refine the design locally, to search for an optimum by splitting cells and to sample non-rectangular regions.
ER -